Question

Find the area of the greatest rectangle that can be inscribed in an ellipse x 2 a 2 + y 2 b2 = 1

Answer 1

2ab is the area of the greatest rectangle that can be inscribed in an ellipse x²/a² + y²/b² = 1

Step-by-step explanation:

refer attached diagram

Area of rectangle = 4xy

A = 4xy

dA/dx = 4xdy/dx  + 4y

x²/a² + y²/b² = 1

=> 2x/a²  + 2y(dy/dx)/b² = 0

=> dy/dx   = - xb²/a²y

dA/dx = 4x( - xb²/a²y)  + 4y

put dA/dx = 0

=> a²y² = b²x²

=> x²/a² = y²/b²

x²/a² + y²/b² = 1

=> x²/a² = y²/b²  = 1/2

x = a/√2  , y = b/√2

Area = 4xy  =  4 ( a/√2)(b/√2)  = 2ab

2ab is the area of the greatest rectangle that can be inscribed in an ellipse x²/a² + y²/b² = 1

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